Finite Group Databasebeta

← C127×#17216 #44580889

Subgroups — C127×#17216 #44580889

Order 32512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393×C127C212yesmaximal, complemented
2C127×#1446C211yesmaximal, characteristic, complemented
3C127×#1442C213yesmaximal, complemented
4C127×#1439C216yesmaximal, complemented
5C127×#1438C213yesmaximal
6C23×Q8×C127C2211yescharacteristic
7C23×D4×C127C2211yescharacteristic
8C22×C4⋊C4×C127C2213yes—
9C22×C22⋊C4×C127C2216yes—
10C22×C22⋊C4×C127C2216yes—
11#393×C127—28no—
12C2×#95×C127C2216yes—
13C2×#95×C127C2213yes—
14C2×#95×C127C2213yes—
15C2×#95×C127C2213yes—
16C2×#95×C127C2213yes—
17C25×C127C2312yes—
18C22×Q8×C127—43no—
19C22×Q8×C127—43no—
20C22×Q8×C127D414yescomplemented
21C22×D4×C127—43no—
22C22×D4×C127—43no—
23C22×D4×C127Q814yescomplemented
24C23×C4×C127C2316yes—
25C23×C4×C127C2313yes—
26C23×C4×C127C2311yescharacteristic
27C23×C4×C127C2313yes—
28C2×C4⋊C4×C127—43no—
29C2×C4⋊C4×C127—46no—
30C2×C22⋊C4×C127—212no—
31C2×C22⋊C4×C127—212no—
32C2×C22⋊C4×C127—46no—
33C2×C22⋊C4×C127—46no—
34C2×C22⋊C4×C127—46no—
35C2×C22⋊C4×C127—46no—
36#95×C127—212no—
37#95×C127—26no—
38#95×C127—43no—
39#95×C127—43no—
40#95×C127—43no—
41C24×C127—22no—
42C24×C127—26no—
43C24×C127—26no—
44C24×C127—26no—
45C24×C127—26no—
46C24×C127—22no—
47C24×C127C2×Q814yes—
48C24×C127C2411yescharacteristic
49C2×Q8×C127—412no—
50C2×Q8×C127—412no—
51C2×Q8×C127—44no—
52C2×D4×C127—412no—
53C2×D4×C127—412no—
54C2×D4×C127—44no—
55C22×C4×C127—26no—
56C22×C4×C127—23no—
57C22×C4×C127—23no—
58C22×C4×C127—23no—
59C22×C4×C127—23no—
60C22×C4×C127—23no—
61C22×C4×C127—26no—
62C22×C4×C127—26no—
63C22×C4×C127—26no—
64C22×C4×C127—23no—
65C22×C4×C127—26no—
66C22×C4×C127—26no—
67C22×C4×C127—26no—
68C22×C4×C127—26no—
69C22×C4×C127—26no—
70C22×C4×C127—26no—
71C22×C4×C127—26no—
72C22×C4×C127C2×D416yes—
73C22×C4×C127C2×Q812yes—
74C22×C4×C127#4116yes—
75C4⋊C4×C127—412no—
76C22⋊C4×C127—424no—
77C22⋊C4×C127—424no—
78C23×C127—212no—
79C23×C127—24no—
80C23×C127—212no—
81C23×C127—46no—
82C23×C127—46no—
83C23×C127—46no—
84C23×C127—46no—
85C23×C127—46no—
86C23×C127—46no—
87C23×C127—46no—
88C23×C127—46no—
89C23×C127—42no—
90C23×C127—46no—
91C23×C127C22×D411yescharacteristic
92C23×C127C22×Q811yescharacteristic
93C23×C127C2×#4113yes—
94C23×C127H(5,𝔽2)13yes—
95C23×C127H(5,𝔽2)13yes—
96C23×C127H(5,𝔽2)13yes—
97C23×C1272−1+411yescharacteristic
98Q8×C127—416no—
99D4×C127—416no—
100C2×C4×C127—212no—
101C2×C4×C127—26no—
102C2×C4×C127—212no—
103C2×C4×C127—26no—
104C2×C4×C127—26no—
105C2×C4×C127—43no—
106C2×C4×C127—46no—
107C2×C4×C127—43no—
108C2×C4×C127—43no—
109C2×C4×C127—43no—
110C2×C4×C127—41no—
111C2×C4×C127—43no—
112C2×C4×C127—46no—
113C2×C4×C127—46no—
114C2×C4×C127—46no—
115C2×C4×C127—46no—
116C2×C4×C127—46no—
117C2×C4×C127—46no—
118C2×C4×C127—46no—
119C2×C4×C127—46no—
120C22×C127—44no—
121C22×C127—412no—
122C22×C127—412no—
123C22×C127—86no—
124C22×C127—86no—
125C22×C127—82no—
126C22×C127—82no—
127C22×C127#53413yes—
128C22×C127#53816yes—
129C22×C127#53913yes—
130C22×C127#54313yes—
131C22×C127D4×Q811yescharacteristic
132C22×C127#55013yes—
133C22×C127#55413yes—
134C22×C127#55513yes—
135C22×C127#56013yes—
136C22×C127H(3,𝔽4)13yes—
137C22×C127H(3,𝔽4)11yescharacteristic
138C22×C127#56213yes—
139C4×C127—46no—
140C4×C127—42no—
141C4×C127—46no—
142C4×C127—86no—
143#17216C12711yesmaximal, characteristic, direct product factor, Sylow
144C2×C127—84no—
145C2×C127#273213yes—
146C2×C127#273313yes—
147C2×C127#278413yes—
148C2×C127#278511yescharacteristic
149C2×C127#281211yescharacteristic
150C2×C127#282413yes—
151C2×C127#284011yescharacteristic
152C2×#393C2×C12712yescomplemented
153#1446C2×C12711yescharacteristic, complemented
154#1442C2×C12713yescomplemented
155#1439C2×C12716yescomplemented
156#1438C2×C12713yes—
157C127#1721611yescharacteristic, direct product factor, Sylow
158C23×Q8C22×C12711yescharacteristic
159C23×D4C22×C12711yescharacteristic
160C22×C4⋊C4C22×C12713yes—
161C22×C22⋊C4C22×C12716yes—
162C22×C22⋊C4C22×C12716yes—
163#393—28no—
164C2×#95C22×C12716yes—
165C2×#95C22×C12713yes—
166C2×#95C22×C12713yes—
167C2×#95C22×C12713yes—
168C2×#95C22×C12713yes—
169C25C23×C12712yes—
170C22×Q8—43no—
171C22×Q8—43no—
172C22×Q8D4×C12714yescomplemented
173C22×D4—43no—
174C22×D4—43no—
175C22×D4Q8×C12714yescomplemented
176C23×C4C23×C12716yes—
177C23×C4C23×C12713yes—
178C23×C4C23×C12711yescharacteristic
179C23×C4C23×C12713yes—
180C2×C4⋊C4—43no—
181C2×C4⋊C4—46no—
182C2×C22⋊C4—212no—
183C2×C22⋊C4—212no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—46no—
188#95—212no—
189#95—26no—
190#95—43no—
191#95—43no—
192#95—43no—
193C24—22no—
194C24—26no—
195C24—26no—
196C24—26no—
197C24—26no—
198C24—22no—
199C24C2×Q8×C12714yes—
200C24C24×C12711yescharacteristic
201C2×Q8—412no—
202C2×Q8—412no—
203C2×Q8—44no—
204C2×D4—412no—
205C2×D4—412no—
206C2×D4—44no—
207C22×C4—26no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—23no—
212C22×C4—23no—
213C22×C4—26no—
214C22×C4—26no—
215C22×C4—26no—
216C22×C4—23no—
217C22×C4—26no—
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—26no—
221C22×C4—26no—
222C22×C4—26no—
223C22×C4—26no—
224C22×C4C2×D4×C12716yes—
225C22×C4C2×Q8×C12712yes—
226C22×C4#41×C12716yes—
227C4⋊C4—412no—
228C22⋊C4—424no—
229C22⋊C4—424no—
230C23—212no—
231C23—24no—
232C23—212no—
233C23—46no—
234C23—46no—
235C23—46no—
236C23—46no—
237C23—46no—
238C23—46no—
239C23—46no—
240C23—46no—
241C23—42no—
242C23—46no—
243C23C22×D4×C12711yescharacteristic
244C23C22×Q8×C12711yescharacteristic
245C23C2×#41×C12713yes—
246C23H(5,𝔽2)×C12713yes—
247C23H(5,𝔽2)×C12713yes—
248C23H(5,𝔽2)×C12713yes—
249C232−1+4×C12711yescharacteristic
250Q8—416no—
251D4—416no—
252C2×C4—212no—
253C2×C4—26no—
254C2×C4—212no—
255C2×C4—26no—
256C2×C4—26no—
257C2×C4—46no—
258C2×C4—43no—
259C2×C4—43no—
260C2×C4—43no—
261C2×C4—43no—
262C2×C4—41no—
263C2×C4—43no—
264C2×C4—46no—
265C2×C4—46no—
266C2×C4—46no—
267C2×C4—46no—
268C2×C4—46no—
269C2×C4—46no—
270C2×C4—46no—
271C2×C4—46no—
272C22—44no—
273C22—412no—
274C22—412no—
275C22—86no—
276C22—86no—
277C22—82no—
278C22—82no—
279C22#534×C12713yes—
280C22#538×C12716yes—
281C22#539×C12713yes—
282C22#543×C12713yes—
283C22D4×Q8×C12711yescharacteristic
284C22#550×C12713yes—
285C22#554×C12713yes—
286C22#555×C12713yes—
287C22#560×C12713yes—
288C22H(3,𝔽4)×C12713yes—
289C22H(3,𝔽4)×C12711yescharacteristic
290C22#562×C12713yes—
291C4—46no—
292C4—42no—
293C4—46no—
294C4—86no—
295C2—84no—
296C2C127×#273213yes—
297C2C127×#273313yes—
298C2C127×#278413yes—
299C2C127×#278511yescharacteristic
300C2C127×#281211yescharacteristic
301C2C127×#282413yes—
302C2C127×#284011yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC23×D4×C127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#393×C127C2semidirect
2. C127×#1446C2semidirect
3. C127×#1442C2semidirect
4. C127×#1439C2semidirect
20. C22×Q8×C127D4semidirect
23. C22×D4×C127Q8semidirect
143. #17216C127direct (N × H)
152. C2×#393C2×C127semidirect
153. #1446C2×C127semidirect
154. #1442C2×C127semidirect
155. #1439C2×C127semidirect
157. C127#17216direct (N × H)
172. C22×Q8D4×C127semidirect
175. C22×D4Q8×C127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#393×C1272
C127×#14462
C127×#14422
C127×#14392
C127×#14382
#17216127